Numerical treatment of stochastic control problems by Fourier-cosine series expansions

Abstract

Financial options are contracts which define rights on stocks in a financial market. Real options arise in for example economical, personal or societal context. The holder has a real option in the sense of a real `choice'. Real options appear in for example the dike height problem, where one has to make optimal choices about when to increase the dike level and by which amount. Another example is the forest harvesting problem, where one can determine the harvesting time. (Real) option prices can often be formulated as a stochastic optimisation problem. The impulse control problem represents a problem class which incorporates jumps on the state process, like an instantaneous jump in the dike level. The dynamic programming principle can be derived from the above representation. The infinitesimal version results in the Hamilton-Jacobi-Bellman (HJB) equation. Two pricing methods for financial and real options have been discussed. The first method is called the dynamic programming approach. In this case we assume that risks cannot be hedged and the holder asks a risk adjusted discount rate for holding the option. In the second approach it is assumed that the asset markets are rich enough to be able to hedge risks and the market will not reward the holder for holding his risky option. This method is called the contingent claims approach. We focused on the dike height problem based on work of Kempker and v.d. Pijl. The state dynamics of a basic model consisted of the dike height and the stochastic economic value of endangered goods, which is exposed to possible floods. The average water level was supposed to follow a deterministic function. Occurrences of extreme water level may cause flooding and they were added to the model. The controller can perform dike heightenings at certain construction costs. A recursive algorithm based on Fourier-cosine series expansions, called the dike-COS method, enabled us to solve the dike height problem. With this we obtained a control law, which describes when it is optimal to increase the dikes and by which amount, depending on the economic value of endangered goods and the current dike level in comparison with the water level. Parameter variation showed that the real option value, in other words the expected costs of flood protection under optimal heightenings, increases in the expected economic growth and the intensity rate of extreme water level and decreases in the risk adjusted discount rate. An alternative model, incorporating a stochastic average water level, was also considered and we developed an extended two-dimensional algorithm. The results showed that the expected costs of flood protection increase if the volatility of the average water level is increased. A method which considers several scenarios of deterministic water level, instead of a stochastic level, may perform satisfactorily to estimate the real option value under uncertain water level rise.